Beyond twisted arcs: a McKay correspondence for reductive groups

TL;DR

This paper introduces warped maps as a generalization of twisted maps, providing a framework for studying Artin stacks and establishing a motivic change of variables formula that leads to a McKay correspondence for reductive groups.

Contribution

It develops warped maps for Artin stacks, proves a valuative criterion for good moduli spaces, and derives a motivic McKay correspondence for reductive groups.

Findings

Canonical lift of arcs in good moduli spaces via warped maps

Framework for relating warped maps to usual maps through an auxiliary stack

Motivic change of variables formula for stacks with applications to McKay correspondence

Abstract

We introduce a natural generalization of twisted maps, called \emph{warped maps}. While twisted maps play an important role in the study of Deligne--Mumford stacks, warped maps are better suited for studying Artin stacks. Heuristically, warped maps see the hidden proper-like behavior satisfied by good moduli space maps. Specifically, we show that every arc of a good moduli space admits a \emph{canonical} lift, in a warped sense, thereby proving a valuative criterion for good moduli spaces. Furthermore, we prove that warped maps to an Artin stack $\mathcal{X}$ are given by usual maps to an auxiliary Artin stack $\mathscr{W}(\mathcal{X})$, immediately obtaining a versatile framework for bootstrapping results about usual maps to the setting of warped maps. As an application we obtain a motivic change of variables formula which, given a stacky resolution of singularities $\mathcal{X} \to Y$, canonically expresses any given motivic integral over arcs of $Y$ as a certain motivic integral over warped arcs of $\mathcal{X}$. In particular, this yields a McKay correspondence for linearly reductive groups.